The metric tensor and the (pseudo-)Riemannian manifolds are defined. The results of the earlier chapters are specialised to this case, in particular the affine connection coefficients are shown to reduce to the Christoffel symbols. The signature of a metric, the timelike, null and spacelike vectors are defined and the notion of a light cone is introduced. It is shown that in two dimensions the notion of curvature agrees with intuition. It is also shown that geodesic lines extremise the interval (i.e. the ‘distance’). Mappings between Riemann spaces are discussed. Conformal curvature (= the Weyl tensor) is defined and it is shown that zero conformal curvature on a manifold of dimension >=4 implies that the metric is proportional to the flat one. Conformal flatness in three dimensions and the Cotton–York tensor are discussed. Embeddings of Riemannian manifolds in Riemannian manifolds of higher dimension are discussed and the Gauss–Codazzi equations derived. The Petrov classification of conformal curvature tensors in four dimensions with signature (+ - - -) is introduced at an elementary level.

Spinors are defined, their basic properties and relation to tensors are derived. The spinor image of the Weyl tensor is derived and it is shown that it is symmetric in all four of its spinor indices. From this, the classification of Weyl tensors equivalent to Petrov’s (by the Penrose method) is derived. The equivalence of these two approaches is proved. The third (Debever’s) method of classification of Weyl tensors is derived, and its equivalence to those of Petrov and Penrose is demonstrated. Extended hints for verifying the calculations (moved to the exercises section) are provided.

Parallel transport of vectors and tensor densities along curves is defined using the covariant derivative. A geodesic is defined as such a curve, along which the tangent vector, when parallely transported, is collinear with the tangent vector defined at the endpoint. Affine parametrisation is introduced.

The curvature tensor is defined via the commutators of second covariant derivatives acting on tensor densities. It is shown that curvature is responsible for the path-dependence of parallel transport. Algebraic and differential identities obeyed by the curvature tensor are derived. The geodesic deviation is defined, and the equation governing it is derived.

Maxwell’s equations in curved spacetime are presented, and Einstein’s equations with electromagnetic field included in the sources are derived. The attempt to unify electromagnetism with gravitation in the Kaluza–Klein theory is presented.

The derivation of the Einstein equations is presented following Einstein’s method. Hilbert’s derivation (from a variational principle) is also presented. The Newtonian limit of Einstein’s theory is discussed. A Bianchi type I solution of Einstein’s equations with a dust source is derived. A brief review of other theories of gravitation (Brans–Dicke, Bergmann–Wagoner, Einstein–Cartan and Rosen) is presented. The matching conditions for different metrics are derived. The weak-field approximation to general relativity is presented.

The Robertson–Walker metrics are presented as the simplest candidates for the models of our observed Universe. The Friedmann solutions of the Einstein equations (which follow when a R–W metric is taken as an ansatz), with and without the cosmological constant, are derived and discussed in detail. The Milne–McCrea Newtonian analogues of the Friedmann models are derived. Horizons in the R–W models are discussed following the classical Rindler paper. The conceptual basis of the inflationary models is critically reviewed.

This chapter contains detailed hints on how to solve the more difficult exercises and verify some other complicated calculations.

The plane- and hyperbolically symmetric counterparts of the L–T models (i.e. the Ellis solutions), and generalisations of all three classes to charged dust source are derived and discussed. It is shown that the most natural interpretation of the plane-symmetric Ellis metric is an expanding or contracting family of 2-dimensional flat tori. The proof of the Ori theorem that for a spherically symmetric weakly charged dust ball shell crossings will block the bounce through the minimal radius is copied in detail. A subcase left out by Ori is discussed, but it will also lead to a shell crossing, only at the other side of the minimal radius. In this special case, a peculiar direction-dependent singularity is present: at the centre the matter density becomes negative for a short period before and after the bounce. The Datt–Ruban solution, its generalisation to charged dust source and the matching of both these solutions to, respectively, the Schwarzschild and Reissner–Nordstr\“{o}m solutions are presented and discussed. In the matched configuration the DR region stays inside the Schwarzschild or RN event horizon.

It is shown how the assumption of symmetry implies the Killing equations (more generally, invariance equations of arbitrary tensors are derived and discussed). It is also shown how to find the symmetry transformations of a manifold given the Killing vectors. The Lie derivative is introduced, and it is shown that the algebra of a symmetry group always has a finite dimension, not larger than n(n+1)/2, where nis the dimension of the manifold. Conformal symmetries are defined and it is shown that the algebra of the conformal symmetry group has dimension not larger than (n+1)(n+2)/2. The metric of a spherically symmetric 4-dimensional manifold is derived from the Killing equations, and its general properties are discussed. Explicit formulae for the conformal symmetries of a flat space of arbitrary dimension are given.

The logical and observational problems of Newton’s theory of gravitation that led Einstein to think about general relativity are briefly presented. In particular, this was the anomalous orbital motion of Mercury and the failed attempts to explain it within Newton’s theory. The local equivalence of inertial and gravitational forces is demonstrated.

The need for differential geometry is explained by considering the construction of parallel straight lines running far from each other in Euclidean space. Generalisation of the notion of parallelism to curved surfaces is explained.

The description of motion of a continuous medium in curved spacetime is introduced and related to the corresponding Newtonian description. Expansion, acceleration, shear and rotation of the medium are defined and interpreted. The Raychaudhuri equation and other evolution equations of hydrodynamical quantities are derived. A simple example of a singularity theorem is presented. Relativistic thermodynamics is introduced and it is shown that a thermodynamical scheme is guaranteed to exist only in such spacetimes that have an at least 2-dimensional symmetry group.

The geometric optics approximation to Maxwell equations is derived. The redshift and the description of bundles of rays via expansion, shear and rotation are defined. Equations of propagation of these optical tensors are derived. The proofs of the Goldberg - Sachs theorem and of the reciprocity theorem are presented. The equations of the Fermi - Walker transport and of the position drift of light sources are derived.